Gaussian curvature and gyroscopes

TL;DR

This paper establishes a connection between Gaussian curvature and gyroscopic forces by analyzing the motion of a spinning disk on a curved surface, providing a new geometric interpretation of gyroscopic phenomena.

Contribution

It introduces a novel mechanical interpretation of Gaussian curvature through gyroscopic forces and relates the precession of Lagrange's top to surface curvature.

Findings

Gyroscopic force equals magnetic force on a charge in a magnetic field normal to the surface.

Precession of Lagrange's top is due to the curvature of a sphere defined by its parameters.

Spin induces a force proportional to Gaussian curvature.

Abstract

We relate Gaussian curvature to the gyroscopic force, thus giving a mechanical interpretation of the former and a geometrical interpretation of the latter. We do so by considering the motion of a spinning disk constrained to be tangent to a curved surface. It is shown that the spin gives rise to a force on the disk which is equal to the magnetic force on a point charge moving in a magnetic field normal to the surface, of magnitude equal to the Gaussian curvature, and of charge equal to the disk's axial spin. In a special case, this demonstrates that the precession of Lagrange's top is due to the curvature of a sphere determined by the parameters of the top.